Theorem srg0cl

Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	srg0cl.b	$$⊢B={\mathrm{Base}}_R$$
	srg0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	srg0cl	$$⊢R\in \mathrm{SRing}\to \underset{\dot{}}{0}\in B$$

Proof

Step	Hyp	Ref	Expression
1		srg0cl.b	$$⊢B={\mathrm{Base}}_R$$
2		srg0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		srgmnd	$$⊢R\in \mathrm{SRing}\to R\in \mathrm{Mnd}$$
4	1 2	mndidcl	$$⊢R\in \mathrm{Mnd}\to \underset{\dot{}}{0}\in B$$
5	3 4	syl	$$⊢R\in \mathrm{SRing}\to \underset{\dot{}}{0}\in B$$